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known to be cont ained in. The shape of this region g(w) is determined by the way in whid the uncertainty is specified. Thus, specification of uncert aunties in amplitude and phase will lead to a sector bound Figure 2. la. Several of the results and met ho ds described in the following are b ased on norm bounded mo del deviations. For SIso systems this means that the uncert ainty at each frequency is bounded by a circle of radius la(w)in the Nyquist plane, see Figure 2. 1b G(w) radius la(w) auncertaintyin amplitude b. norm bounded uncertainty and phas Figure 2.1: Uncertainty regions by spec fication of(a) amplitude and phase uncertainty (6) norm bounded uncertainty Based on this, a family of models can be defined c={G△:|G△()-G(j)≤Ca(u)} (21) {G△:|G△(元)-G(ju)≤Cm(u)G(j) (22) where G(u denote the nominal model, and Ga(u) denote possilbe models of the process ea(w) is the maximal additive model uncert ainty, and em(w) is the maximal multiplicative (relative)uncertainty. An arbitrary member of g can be described as G△(ju)=G(ju)(1+△m(u)=G(u)+△a(ju) where the actual multiplicative or additive model deviation(Am or Aa) is bounded by 1△m(ju)≤nm(u) (24) △a(ju)≤a(a) (25) A multiplicative model uncert ainty description often increase with increasing frequences the models applied for compensator design are derived with emphasis on the description of the dominating dynamics If em(w)> 1, the norm bounded model deviation allow the mo dels to have a different number of zeros in the right half plane. A zero on the jw axis is faciliated e.g. by Am gw)=-1, ) 23 ' ) ''( / ) 23 a. uncertainty in amplitude and phase b. norm bounded uncertainty ) Re Im Im Re G(jw g(w) g(w) radius la (w) G(jw) ) 23! 4 ! $23% $22% $ % " ! $2.% $ % ! $2-% $2,% " 5 6 " 6 $2.%